A spherical aluminum ball of mass 1.26 kg contains an empty spherical cavity that is concentric

step 1 So this is the water surface and the ball floats like this where this is air with radius R. This

A spherical aluminum ball of mass 1.26 kg contains an empty...

A spherical aluminum ball of mass 1.26 $\mathrm{kg}$ contains an empty spherical cavity that is concentric with the ball. The ball barely floats in water. Calculate (a) the outer radius of the ball and (b) the radius of the cavity.

Key Concepts

Spherical Geometry

The geometry of spheres is governed by the formula for volume, (4/3)?r³. This relationship is crucial when connecting the radii of the sphere and any internal features (like a cavity) to their respective volumes, which in turn are used to link mass, density, and buoyancy in problems involving spherical objects.

Composite Bodies

A composite body is one that comprises materials with different densities or includes voids, such as cavities. Analyzing composite bodies involves partitioning the total volume among the different regions (solid material and cavity) to determine how the varying densities contribute to the overall buoyancy or stability of the object.

Archimedes’ Principle

This principle states that the buoyant force on an object immersed in a fluid equals the weight of the fluid displaced by the object. It is critical when analyzing floating bodies, as the condition of buoyant equilibrium (weight equals displaced fluid weight) allows one to determine the object's overall volume based on its mass and the fluid's density.

Density

Density, defined as mass per unit volume, is a key property of materials. Comparing the density of an object with that of the surrounding fluid determines whether the object sinks, floats, or remains neutrally buoyant. In problems involving composite bodies, the overall density is calculated by considering the masses and volumes of the available components.

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